Here's a problem on producing a counter example that I'm having difficulty with. Any help/hint would be much appreciated. Thank you.
The Problem : To produce a function $f$ for which the following limit exists $$\lim_{h \to 0}\frac{f(x+h)-f(x-h)}{2h}$$ But $f$ is NOT differentiable at $x$. $($i.e. we are trying to contradict the false converse to the correct statement that if $f$ is differentiable at $x,$ then the said limit exists and is equal to $f'(x),$ by producing a counter example$)$.
On a general note, producing counter examples is one of my weaknesses. Do you just remember counter examples or there are ways to actually construct them using logical steps? Any general tips would also be much appreciated.